Question

Prove that $\mathcal{F}_{m}$ has $2^{2^{m}}$ elements.

   Prove that $\mathcal{F}_{m}$ has $2^{2^{m}}$ elements.

Measure, Integral and Probability

Measure, Integral and Probability

Marek Capi?ski,… 2nd Edition

Chapter 2, Problem 12

Instant Answer

Solved by Expert James Chok

10/18/2019

Step 1

First, recall that $\mathcal{F}_{m}$ is the set of all functions from a set $A$ with $m$ elements to a set $B$ with 2 elements. In other words, $\mathcal{F}_{m}$ is the set of all functions $f: A \to B$ where $|A| = m$ and $|B| = 2$. Show more…

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Prove that $\mathcal{F}_{m}$ has $2^{2^{m}}$ elements.

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Key Concepts

Counting Principle for Functions

In combinatorics, the number of functions from one finite set to another is determined by raising the number of choices for the output to the power of the number of inputs. For a function defined on a domain of size n with each element having k possible outputs, the total number of such functions is k^n.

Exponential Growth in Combinatorics

The problem demonstrates double exponential growth: when the domain itself grows exponentially (as with 2^m input combinations for m binary variables), and for each input two outputs are possible, the total number of distinct functions becomes 2 raised to the power of 2^m. This concept illustrates how quickly the number of possibilities can escalate in combinatorial settings.

Boolean Function

A Boolean function is a mapping that takes a fixed number of binary inputs (each input being either 0 or 1) and produces a binary output. The function is completely described by its outputs on all possible input combinations, making it a fundamental object in discrete mathematics and computer science.

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